The complement bimetric-dimension of corona graphs

Let G be a connected graph, V (G) be a vertex set of G, d(u, v) be a length of shortest path between u, v ∈ V(G), and δ(u, v) be a length of longest path between u, v ∈ V(G). For every vertex u and an ordered subset S = {s1, s2, s3,.., sk} ⊆ V (G), let a and b of (a, b) be a pair k-tuple that is a = (d(u, s1), d (u, s2),.., d (u, sk)) and b = (δ (u, s1),δ (u, s2),.., δ(u, sk)). It has been stated that the subset S is called a bimetric-generator set of G if every two different vertices in G have distinct bimetric-representation with respect to S, and if cardinality of S is minimum then S is called bimetric-basis of G and its cardinality denoted by βb (G) that is bimetric-dimension of G. To develop the concept of bimetric-dimension, this research introduces a concept of complement bi-metric dimension. Subset S is called a complement bimetric-generator set of G if at least there are two vertices in G have the same complement bimetric representation with respect to S. If the cardinality of the such S is maximum then S called as complement bimetric basis of G and its number of vertices is complement bimetric-dimension of G, denoted by βb¯. It is also discussed in this research about a complement bimetric-dimension of corona product graphs. The complement bimetric-dimension of corona product graphs of two connected graphs G and H, βb¯(G ⊙ H), is βb¯(G ⊙ H) = (|V (G) - 1)(|V (H)| + 1) + βb¯(K1 + H), is influenced by the order of each graph and the complement bimetric-dimension of K1+H.